Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $x = \dfrac{3(2a + 5)}{-10} \div \dfrac{3(2a + 5)}{4} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{3(2a + 5)}{-10} \times \dfrac{4}{3(2a + 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 3(2a + 5) \times 4 } { -10 \times 3(2a + 5) } $ $ x = \dfrac{12(2a + 5)}{-30(2a + 5)} $ We can cancel the $2a + 5$ so long as $2a + 5 \neq 0$ Therefore $a \neq -\dfrac{5}{2}$ $x = \dfrac{12 \cancel{(2a + 5})}{-30 \cancel{(2a + 5)}} = -\dfrac{12}{30} = -\dfrac{2}{5} $